Ph.D. thesis (pdf) - dirac

134 Mean squared displacement
I P is of the order of magnitude 4 for fragile liquids at T g (τ = 100s) while T g α P ∼ 0.1,
meaning that the last term can be neglected. Using the general relation between the
conventional steepness index and the Olsen index (equation 2.2.6) it subsequently
follows that the model predicts a proportionality between fragility and the relative
change of 〈u 2 〉 with relative change in temperature:
m P
( )
( )
τg
τg ∂ log〈u 2 〉
= log 10 (1 + I P ) = log
τ 10 0 τ 0 ∂ log T ∣ (7.5.5)
P
= 16 ∂ log〈u2 〉
∂ log T ∣ . (7.5.6)
P
The last equality follows if all values are evaluated at T g defined by τ g = 100 s and if
it is assumed that τ 0 = 10 −14 s. Hence the elastic model predicts a correspondence
between the slope seen in figure 7.13 and the fragility found from the temperature
dependence of the alpha relaxation time.
Figure 7.14 tests this relation, using fragilities and T g ’s taken from literature (see
appendix for values and references). We also include some ∂ log〈u2 〉
∂ log T
∣ (T = T g )
P
calculated on the basis of mean square displacements reported in literature. The
value of ∂ log〈u2 〉
∂ log T
∣ (T = T g ) is in all cases calculated in a narrow range temperature
P
range from T g to ∼ 1.1T g , because this corresponds to the range where fragility is
determined. It has to be stressed when considering this figure that the elastic model
not only predicts a proportionality between m P and ∂ log〈u2 〉
∂ log T
∣ , the elastic model
P
predicts the proportionality constant as well, hence the line is not a fit nor a guide
to the eye. The line appearing in the figure is a parameter free prediction of the
elastic model. It is therefore quite striking and not at all trivial, that the order of
magnitude is correct. Secondly it also appears that the variations in m P follow the
variations in ∂ log〈u2 〉
∂ log T
∣ except for the very fragile liquids.
P
A further test of the predicted correlation, could in principle be to consider the pressure
dependence of fragility and ∂ log〈u2 〉
∂ log T
∣ . This correlation is of the type discussed
P
in section 3.4, that is a correlation between fragility and the temperature dependence
of another quantity. This means that both quantities are path dependent, and the
∣ (T = T g ) is hence expected to follow the pressure dependence of
P
isobaric ∂ log〈u2 〉
∂ log T
the isobaric fragility. The isobaric fragility of DBP is not pressure dependent in the
relevant range. It is therefore consistent that we find that the whole 〈u 2 〉 temperature
dependence collapses after scaling with ρ (−2/3)
g and T g (P) (figure 7.10). The
isobaric fragility of cumene increases with pressure. This means that the correlation
implies that the slope of 〈u 2 〉 should be lower above T g . Figure 7.11 does not
support this prediction, it could even be argued that it contradicts it. However, the